Research Article Mechanism and Stability Analysis of Deformation Failure of a Slope

Hindawi Advances in Civil Engineering Volume 2021, Article ID 8949846, 16 pages https://doi.org/10.1155/2021/8949846

Yingfa Lu ,1 Gan Liu,1 Kai Cui,2 and Jie Zheng3

1School of Civil Engineering and Resource Environment, Hubei University of Technology, Wuhan, Hubei, China 2Key Laboratory of High-Speed Railway Engineering of the Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan, China 3Zhengzhou Highway Development Center, Zhengzhou, Henan, China

Correspondence should be addressed to Yingfa Lu; [email protected]

Received 19 May 2021; Revised 19 June 2021; Accepted 31 July 2021; Published 27 August 2021

Academic Editor: Guang-Liang Feng

Copyright © 2021 Yingfa Lu et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Force distribution during progressive slope failure is an important element in slope stability analysis. In this study, five mechanical failure modes are proposed for thrust- and pull-type slopes, respectively, and five field forms of thrust-type slopes are described. +e properties of progressive failure are evaluated quantitatively: the failure mode of slope obeys the geo-material rule under the peak stress state, and the instability range is gradually developed. +e critical stress state zone is in the process of dynamic change with the development of deformation. It appears that the driving sliding force is greater than the frictional resistance along the sliding surface. When rock or soil stabilizing stresses are at maximum, the vector sum of the driving sliding stress and stabilizing stress is equal to zero at the critical state. +e frictional resistance is equal to the driving sliding force in the stable and less-stable regions, and the normal pressure is wherever equal to the counterpressure. Rigid, flexible, and rigid-flexible design theories are proposed for slope control. New terms are defined and used to evaluate the stability. +e conventional local and surplus stability factors of slopes and their calculation are explained. +e force distribution rule is analyzed during progressive failure, and the conventional stability factor definition is discussed. +e geological settings and monitoring data of landslides are used to analyse changes in the critical stress state. An example is given to illustrate the failure process analysis. +e results show that progressive failure can be well represented and the safety factor can be well described by the main thrust method (MTM), comprehensive displacement method (CDM), and surplus displacement method (SDM), which can be used to feasibly evaluate slope stability.

1. Introduction to be statically analyzed, different limit equilibrium slice methods have different assumptions about the action +e slope failure mechanism and stability analysis are point of the force on the bottom edge of the slice and the traditional topics in geotechnical engineering. Geo- direction and action line of thrust between the slices. technical disasters such as landslide, collapse, rockburst, With the development of numerical analysis, more and and water inrush have occurred frequently in recent more scholars have begun to try other calculation years [1, 2]. More than ten sort of limit equilibrium methods, such as chart-based slope stability assessment stability calculation methods for slope were given in using the generalized Hoek–Brown criterion and ex- previous studies, such as the Fellenius method [3], the tremum solutions to the limit equilibrium method simplified Bishop method [4], the Spencer method [5], subjected to physical admissibility [19–31]. Other re- the Janbu method [6], the Sarma method [7], the wedge searchers established a three-dimensional strict equi- method, and the finite element strength reduction librium equation based on certain assumptions and method (SRM) [8–11]. In traditional slope stability an- proposed four criteria: potential sliding surface mini- alyses, the limit equilibrium slice method is the most mum parameter value criterion, slope active reinforce- often used [12–18]. For the slope with a given slip surface ment force upper limit criterion, lower limit criterion for 2 Advances in Civil Engineering sliding surface, and sliding and upper bound criterion 2.1. ,rust-Type Landslide. +e deformation and force for potential sliding surface selection [32]. transfer of landslides are established based on the fundamental +e finite element method is the most widely used mechanical behaviours of geomaterials. +e load-displacement numerical analysis method in geotechnical engineering. In characteristics of rock and soil can be presented as types I and the field of slope analysis, the main software packages with III curves (Figure 1); any point on the sliding surface expe- strong functionality and wide application are ABAQUS, riences elastic stress, elastoplastic stress, critical stress, post- ANASYS, GeoStudio, rational infiltration analysis, etc. +e failure stress, and residual stress states from the initiation to the corresponding analysis method is based on small defor- mechanical failure of the slope. At a certain moment, points mation assumption and is usually applied to continuum a1, a2, a3, and a4 correspond to the postfailure stress state; mechanics media. Other numerical methods such as the fast point a4 to the critical stress; points a5 and a6 to the elasto- Lagrangian analysis of continua (FLAC) method, discrete plastic stress state; and point a7 to the elastic stress state in a 2D element method, discontinuous deformation analysis landslide problem. +e postfailure stress state can be defined as (DDA) method, and popular element method have been an unstable zone, the critical stress state as a critical zone, the increasingly applied to landslide stability analysis. elastoplastic stress state as a less-stable zone, and the elastic +e above landslide stability analysis is based on the limit stress state as a stable zone (Figure 1). equilibrium state. However, landslide damage is a process of +e failure occurs along the soft interlayer gradual development [33–40]. +at is, some zones are in the (a1, ... , a5, ... , a8), which is defined as mode I. In the rear postfailure stress state, some zones are in the critical stress part, the failure occurs along the weak intercalation state, some zones are in the prepeak stress state [41, 42], and (a1, ... , a5) and along the sliding body (a5, a9) in the front some zones are in the small deformation state, and this part of the landslide, which is classified as mode II. +e evolution process changes with changes in the environment. failure modes are controlled by the mechanical behaviours of However, the traditional method does not consider the soils/rocks. nonuniform distribution of the driving force along the sliding surface; different numerical analysis methods have 2.2. Pull-Type Landslide. At a certain moment, points certain difficulties in dealing with the discontinuous surface. (b , b ) correspond to the postfailure stress state and point +erefore, this paper proposes a new analytical method to 1 2 (b ) corresponds to the critical stress state. Points (b , b , b solve the above problems. 3 4 5 6 or b , b , b ) are before the peak stress, and the whole sliding In this paper, the sliding surface is divided into an 7 8 9 surface can be classified into unstable, critical, less-stable, unstable zone, a critical zone, a less-stable zone, and a stable and stable zones (Figure 2). +e failure occurs along the soft zone. +e transfer law of the landslide force is analyzed, and interlayer (b , ... , b ), which is defined as mode I. In the the characteristics of the critical block (or unit) force of the 1 6 front part, the failure occurs along the weak intercalation slope are proposed. +e failure mechanisms, natures, and (b , b , b ) and failure occurs along the sliding body destruction control standards of thrust-type, pull-type, and 1 2 3 (b , b , b ) in the rear part of the landslide, which is classified mixed landslides are described. Based on the deformation 7 8 9 as mode II. In the front part, failure occurs along the weak and stress analysis, combined with the possible failure modes intercalation (b , b , b ) and along the section of the sliding of slope, the following methods are defined: comprehensive 1 2 3 body (b , b ) as traction strength control, which is defined sliding-resistance, main thrust, comprehensive displace- 3 12 as mode III. +e failure modes are also controlled by the ment, and surplus displacement methods. +e above analysis mechanical behaviours of soils/rocks. methods can be used to analyse slope progressive failure stability and provide a new idea for the study of landslide stability. 2.3. Mixed Failure Mode. +e thrust- and pull-types are the main failure modes during the progressive deformation process 2. Failure Modes of landslides. However, a mixed failure mode of thrust- and pull-types can take place. For instance, tensile-shear failure Landslides occur due to long-term geological processes, occurs at depth (see Figure 3, point c4), while the tensile stress environmental factors, human engineering, etc. Few studies increases to its strength. Traction failure (σn) occurs along have focused on the progressive deformation process from section (c4c8). +e first scarp is formed. If the tensile strength the initiation crack to the failure of landslide. In this paper, does not reach its strength, this scarp does not develop, and a the whole failure process of the landslide can be described by pull-type landslide is certainly produced in zone the different stress distribution characteristics of sliding (c4c8c9c1c2c3c4). When the tensile stress reaches the tensile surface points. Only one point along the sliding surface is in strength value, the second scarp (c9c3) forms; the segment the critical state when the sliding force is equal to the sliding (c5c4c3) is in a postfailure stress state, and a corresponding resistance for a 2D landslide, and this stress distribution is unbalanced (or driving sliding) force is shown. +e sliding defined as the critical state. +e remaining points are in the body (c3c4c5c6c7c8c9c3) exhibits thrust-type characteristics, postfailure stress state, and their sliding force is greater than and the driving sliding force impels the sliding body ahead. the antisliding force; the landslide is in a state of “mechanical Two unstable, critical, less-stable, and stable zones appear failure” at this moment. In the following, the deformation during this period of landslide deformation. When the critical mechanism, failure modes, and destruction control stan- state disappears in the c9c1c2c3c9 zone, the thrust-type feature dards are analyzed from a mechanical viewpoint. is shown for the whole landslide (Figure 3). Advances in Civil Engineering 3

τ development of progressive deformation, and the mechanical Type III curve failure properties of the two diﬀerent types of slopes are analyzed.

τpeak Less stable zone-type I 3.1. Characteristics of the ,rust-Type Slope. A continuous

τyield shear failure occurs from the rear (see Figure 5, point A) to the front (see Figure 5, point C) (or from point G to Q of the soft τresid

Stable zone-type III zone-type Stable interlayer) regions of a thrust-type slope during the develop- Stable zone-type I Type I curve ment of progressive deformation. +e stress at the failure point

Unstable zone-type I zone-type Unstable is in the peak state; the elastoplastic stress state is located in γyield γpeak γresid γ front of the failure point; and the postfailure stress state is behind the failure point (Figure 5). +is failure mode is deﬁned as mode I, and a shear failure occurs along the entire sliding surface. Tensile or tensile-shear failure occurs in the rear zone (see Figure 5, DB), and shear failure occurs in the front zone a 1 (see Figure 5, BC). +en, possible shear failure occurs in the ABD triangular area, which can be deﬁned as mode II. In mode a III, shear failure occurs in the rear zone (see Figure 5, ABE) and 2 tensile (or tensile-shear) failure occurs in the front zone (see a 3 Figure 5, EF). Mode IV is a combination of modes II and III, in a a which failure occurs in the rear zone by tensile (or tensile- 9 a 4 a 5 a 6 shear) failure, in the middle zone by shear failure, and in the 8 a 7 front zone by tensile (or tensile-shear) failure (see Figure 5,

critical state less stable zone DBEF). Mode V corresponds to a rock mass with distributed stable zone unstable zone joints (or fissures), and shear, tensile, and tensile-shear failures occur along the soft interlayer and joints (or fissures) (see Figure 1: Failure modes of thrust-type landslides. Figure 5, GHIJ...KLMP) alternately. +e stability classification (stable zone, less-stable zone, critical zone, and unstable zone;

At a certain moment, points (b1, b2) correspond to the Figure 1) along the entire sliding surface applies for the five postfailure stress state and point (b3) corresponds to the failure modes of a thrust-type slope. critical stress state; points (b4, b5, b6 or b7, b8, b9) are before the peak stress, and the whole sliding face can be classified into unstable, critical, less-stable, and stable zones (Figure 2). 3.2. Characteristics of a Pull-Type Slope. Shear failure occurs from the front (see Figure 6, point B) to the rear (see Fig- +e failure occurs along the soft interlayer (b1, ... , b6), which is defined as mode I. In the front part, failure occurs ure 6, point A) regions of a pull-type slope during the de- along the weak intercalation (b , b , b ), and failure occurs velopment of progressive deformation. +e stress at the 1 2 3 failure point is in the peak state; the elastoplastic stress state along the sliding body (b7, b8, b9) in the rear. +e second mixed failure mode is often involved in a is behind the failure point; and the postfailure stress state is landslide with progressive deformation. At first, the landslide in front of the failure point. +is failure mode is defined as initiates in the rear and front parts of the landslide body along mode I. Tensile or tensile-shear failure occurs in the rear the sliding face, and the postfailure stress state corresponds to zone (see Figure 6, DC), and shear failure occurs in the front zone (see Figure 6, CB). +en, possible shear failure occurs in the rear and front parts. A thrust- (d1d2 zone) and pull- (d6d5 zone) type landslide are present in the rear and front parts on the ACD triangular area, which can be defined as mode II. In the sliding surface, respectively. An antisliding feature takes mode III, shear failure occurs in the front zone and tensile failure occurs in the rear zone of the sliding body (see place in the middle part (d2d3d4d5 zone). Two unstable, critical, less-stable, and stable zones appear during the pro- Figure 6, EF). Mode IV corresponds to a rock mass with gressive deformation process (Figure 4). With time and the distributed joints (or fissures): shear, tensile, and shear development of deformation, the antisliding area decreases and failures occur along the soft interlayer and joints (or fissures) ... the failure area increases. Finally, only one point corresponds to alternately (see Figure 6, GHIJ KLMP). Mode V corre- the critical state, and the whole landslide is at the point of sponds to a combination of mode IV and a shear failure (or “mechanical failure.” +e landslide initiates at the front, rear, or tensile-shear or tensile failure (see Figure 6, ET, or ES, or middle part of the landslide along the sliding face, which can be EF)) of the sliding body. +e previous peak stress state is defined the “mixed failure mode.” behind the zone, and the postfailure state is in front of the zone corresponding to a critical state for the five mechanical failure modes of the pull-type slope. 3. Failure Process Analysis +e failure mode of the slope on the sliding surface point obeys 3.3. Characteristics of Deformation. Generally, a slope the mechanical rule of rock and soil mass during the consists of a sliding body, sliding surface, and landslip bed. 4 Advances in Civil Engineering

(a) b9 b6

b 5 b8

b b4 7

b12

b11

b2 b10

(b) τ

b7′

b b9′ 8′

b4′ b5′ b3′

b 6′ b1′

b2′ O γ critical state less stable zone stable zone unstable zone

Figure 2: Failure modes of pull-type landslides.

c c1 9 d1 c8 c2 d2 c3 d3 c4

c5 d4

c6 d5

critical state less stable zone stable zone unstable zone d6 Figure 3: Failure mode of initiation in the middle part of the sliding face. critical state less stable zone stable zone unstable zone

Figure 4: Failure of initiation on the front and rear part of the +e values of the shear and tensile strengths of the sliding sliding face. surface, sliding body, and landslip bed follow a small, moderate, and large pattern. At first, the failure of the sliding surface occurs and the deformation of the sliding body elastoplastic, peak, postfailure, and residual stress states are produces a sliding surface. +e stress state of the sliding shown by each point on the sliding surface during the surface changes with the development of deformation. +e progressive failure process. +e five different stress states characteristics of the complete stress-strain process are may be represented by the points on the sliding surface at the presented by each point on the sliding surface. +e elastic, same time. +e driving sliding force on the sliding surface is Advances in Civil Engineering 5

G A D H B I E J F C

Q K

L M

Figure 5: +e ﬁve failure modes of a thrust-type slope.

P A D M T L S α K 2 F C α 1

B J I H

Figure 6: +e five failure modes of a pull-type slope. greater than the frictional resistance in the postfailure zone. postfailure and residual stress states. In form IV, the entire Certainly, an unbalanced thrust exists for a slope along the sliding surface is in the postfailure and residual stress states. sliding surface. In the front zone of the sliding surface, the In form V, only the residual stress state is present for the driving sliding force is equal to the frictional resistance on entire sliding surface (Figure 7). Form I cannot significantly the sliding surface and a surplus frictional resistance cor- perform slope control without additional reinforcement. +e responding to the strength existing for the thrust-type slope. critical state is often the peak stress state for form II, and +rust-type slopes in the field exhibit five forms. In form I, whether the slope controls are performed depends on the the elastic stress state covers the entire sliding surface. In human influence. +e critical state does not necessarily form II, the whole sliding surface is in the stress state before correspond to the peak stress for forms III, IV, and V, but the the peak stress. In form III, a zone of the sliding surface is in critical state exists for thrust-type slopes. Large deformation the previous peak stress state, another zone is in the critical is produced for these three slopes. +e geometric shape after stress state, and the remainder of the sliding surface is in the deformation may possibly be favourable to thrust-type slope 6 Advances in Civil Engineering stability, but performing slope control is necessary if the +e moment balance equations in the XY-, YZ-, and ZX- effects on humans are important. planes of the critical state must exist for the slice block method under failure conditions:

4. Force Distribution on the Sliding Surface � Mxy � 0,

In this section, the different failure modes are introduced � Myz � 0, (3) during the progressive deformation process and the force � M � . distribution properties on the sliding surface are explained zx 0 taking the thrust-type slope as an example. By considering the field state, the combination of An element (Figure 8) considered to represent the stress equations (2) and (3) may be chosen for the slice block state of the sliding surface, where σu, σu, and σu are the n τ θ method. For instance, the moment balance equation is stresses from the sliding force of the sliding body and satisfied in the XY-plane and the force balance equations σ b, σ b, and σ b are the frictional stresses from the landslip n τ θ exist in the X-, Y-, and Z-axial directions at the critical state bed. +e driving shear stresses are greater than the frictional for the slice block method: shear stresses in the postfailure zone along the sliding surface. +e frictional shear stress reaches a maximum, and XY − plane : � Mxy � 0, the driving shear stresses are equal to the frictional shear stresses at the critical state. +e frictional shear stresses are X − axial direction : � Fx � 0, (4) equal to the driving shear stress in the front zone, corre- Y − axial direction : � Fy � 0, sponding to the critical state. But, the frictional shear stresses do not reach their maximum; the normal stress is equal to Z − axial direction : � Fz � 0. the counterstress in the whole sliding surface. +e shear displacement is discontinuous in the postfailure zone; both When tensile and shear failures occur for the same slice shear stress and shear strain in the postfailure zone on the block, the tensile and shear stresses must be equal to their sliding surface are clearly discontinuous and present a strength values. However, the shear failure has been studied challenge for numerical analysis. Certainly, the driving only for widely used methods, such as the simplified Bishop sliding force (Pi) and pressure (Ni) result from the sliding method, Janbu method, Sarma method, Morgenstern f bed; the frictional force (Fi) and counterpressure (Ni ) method, and SRM. +e tensile failure is negative. proceed from the landslip bed (Figure 9) for a two-di- +e failure development of a slope can be prevented by mensional thrust-type slope. +e driving sliding force is an antislide tie, which can be explained by the critical state greater than the frictional resistance in the postfailure zone based on equations (1)∼(3), and the slope control position on the sliding surface, and the driving sliding force is equal and safety factor can be redefined as follows. to the frictional resistance in the other zones. However, the +e first method: the position of slope control is chosen maximum frictional resistance is reached at the critical state, at the critical state for thrust- and pull-type slopes and at the when the pressure is equal to the counterpressure on the yield limit stress state for the foundation pit; a rigid design is whole sliding surface. used with a safety factor, and a small deformation is per- mitted for this rigid design. +e second method: the position of slope control is chosen at the yield limit stress state for 5. Critical State thrust- and pull-type slopes and at the peak stress state for the foundation pit; a flexible design is applied with a safety +e slope stability is related to the critical state. +e critical factor, and a large deformation is allowable for this flexible state is an obstacle to unstable zone development. +ree design. +e third method: the position of slope control is stress balance equations exist at the critical state for nu- chosen between the first and the second positions, and this merical analysis on the sliding surface: � � � � design can be called a “rigid-flexible” design. +e force and � u� � b� �σθ � ��σθ�, moment balance in equations (1)∼(4) must be satisfied at the � � � � design position with a safety factor, the antisliding force is � u� � b� �στ � ��στ�, (1) the vector sum of the unbalanced thrust from the postfailure � � � � zone, and the displacement of the slope control can be � u� �� b�. σn �σn� calculated. +e force balance equations in the X-, Y-, and Z-axial directions at the critical state must be satisfied for the slice 6. Shear Stress-Strain Model block method under the shear failure condition: +e complete process shear-strain model (CPSM) must necessarily be employed to describe the progressive failure � Fx � 0, process of a slope. A stress-strain equation with four pa- � Fy � 0, (2) rameters is a part of the CPSM. +e equation can be described by taking a shear stress and strain as an example in � F � . z 0 the following form: Advances in Civil Engineering 7

τ τ τ e 3 e2 e a c c c c 4 a 3 4 3 2 1 4 a c5 a 1 e5 e1 a5 2 γ γ γ

b1 b2 f b3 d 1 b4 Stable zone 1 Unstable zone b5 d f2 d 2 Less stable zone Critical stress state d 3 f d 4 3 5 f4 Less stable zone Stable zone f 5 Stable zone

(a) (b) (c) τ τ g g4 5 i 5 i4 g3 i3 g g i i1 2 1 γ 2 γ

h 1 j1 j h2 Unstable zone 2 h3 j3 h j4 h5 4 j 5 Unstable zone

(d) (e)

Figure 7: Five forms of the slope existing in the ﬁeld.

1 P σn 1 N f 1 Unstable zone 1 σθ σ 1 u 3 N τ στ 3 στ i … σθ 3 F1 σ u σn θ u … σ Pi n Ni+1 4 4 2 σ σθ 2 σθ N f τ σ b στ i n b Pi+1 4 σθ 2 Fi σn b σ στ n Stable zone Nm f … Ni+1 P Fi+1 NL m … N Critical stress state Figure 8: Schematic of a sliding surface element. L+1 P … N L F N f n … m m PL+1 Less stable zone Pn . . q ξ f 1 + c FL N � Gc� � , (5) . . L τ F p L+1 N f Fn L+1 f where τ and c are the shear stress and shear strain, re- Nn G spectively; is the shear modulus dependent on the normal Figure stress; and p, q, and ξ are the constant coefficients dependent 9: Distribution characteristics of force along the sliding surface of a 2D thrust-type slope. on the normal stress. +e units of τ and G are kPa. +e following conditions are needed for the rock or soil with softening mechanical behaviours: where cpeak is the critical shear strain corresponding to the 1 + qξ ≠ 0, critical shear stress. (6) +e Mohr–Coulomb criterion is assumed to describe the − 1 < ξ ≤ 0. critical shear stress (τpeak) (note: other criteria can also describe the critical shear stress): +e critical shear strain (defined as the shear strain corresponding to the peak shear stress) is satisfied in the τpeak � C + σn tan φ, (8) following form: q where C is the cohesion, σn is the normal stress, and φ is the p +(1 + qξ)c � 0, ( ) peak 7 frictional angle. +e units of σn and C are kPa. 8 Advances in Civil Engineering

+e critical shear strain is assumed to be related only to z normal stress, and the critical shear strain (cpeak) can be described as follows:

A c 2 ζN peak σn − a2 � B � � +� � � 1, (9) Dy Ex a3 a1 D E

Ω f where a1, a2, a3, and ζN are the constant coefficients which Ω f x a a y are dependent on the normal stress; the units of 1 and 2 are R Ωy R kPa, and a and ζN are the constant coefficients without Ωx 3 Ey Dx units. Cy Cx Finally, C 2 G � G0 + b1σn + b2σn, (10) where G is the initial shear modulus when the normal stress D f 0 z Ωz x Ez (σn) is equal to zero, b1 is a constant coefficient without units, R y and b2 � − b1/(2a2). Ωz +e softening coefficient (ξ) can be presented in the Cx following form: Figure 10: Distribution of failure zones and projections along the X-, Y-, and Z-axes. ξ0 ξ � c ς , (11) 1 +ξ /ξc − 1 �σn/σ � � 0 n P � B σu + σu �dydz, xs f R θ τ Ωx +Ωx where ξ0 is the value of ξ when σn is equal to zero, ξc is the c P � B u + u � x z, value of ξ when σn is equal to σn , and ς is a constant co- ys f σθ στ d d (12) + R efficient without units. +e softening coefficient can be Ωy Ωy u u obtained by the shear stress and shear strain complete Pzs � B σθ + στ �dxdy, f+ R process tests with different normal stresses. Ωz Ωz Regarding the physical significance, it is not neces- where Pxs,Pys, and Pzs are the vector sums of the sliding sary to explain the parameters related to the forces in the directions of the X-, Y-, and Z-axes, Ω is the area f R Mohr–Coulomb criterion: a2 is the critical normal stress of the whole integration, Ω is the destroyed region, and Ω crit crit (a2 � σn ), a3 is the critical shear strain (a3 � cpeak) when is the nondestroyed region. the normal stress is equal to the critical normal stress +e vector sum (Ps) of Pxs,Pys, and Pzs is a � crit a �� ( 2 σn ),��1 is relative to the critical shear strain 2 0 2 P � P �2 +�P � +P �2. (13) (cpeak � a3 1 − (a2/a1) ) when the normal stress is equal s xs ys zs to zero, a1 > a2ξ represents the softening degree of rock or soil under different normal stresses, and p and q are the +e direction cosines of Ps with Cartesian coordinate axes are αs, βs, and cs. +e failure mode can be analyzed, the dis- representative parameters between the critical shear p,b p,b p,b stress and shear strain. tribution of skid-resistance stresses (σn , σθ , and στ ) under the possible failure mode can be obtained, and the vector sums of the antislip force under the possible failure modes can be cal- 7. Stability Analysis culated in the directions of X-, Y-, and Z-axes:

p,b In stability analysis, the mechanical behaviours of weak TxT � B �σp,b + σ + σp,b�dydz, f R n θ z intercalated strata are key factors for the sliding body and Ωx +Ωx p,b diﬀerent safety factors are proposed to describe the slope TyT � B �σp,b + σ + σp,b�dxdz, f R n θ z (14) stability. Ωy +Ωy p,b TzT � B �σp,b + σ + σp,b�dxdy, f R n θ z Ωy +Ωy 7.1. Stability Analysis of an Ideal Elastoplastic Model where TxT,TyT, and TzT are the vector sums of the stabi- 7.1.1. Comprehensive Sliding-Resistance Method (CSRM). lizing forces in the directions of the X-, Y-, and Z-axes under +e stress ﬁelds of the landslide body along the slip surface the possible failure modes. (see Figure 10, dashed line ABDEC represents the sliding +e vector sum (TT) of TxT,TyT, and TzT is surface) can be obtained by the current calculation method, �� 2 2 2 and the vector sums of sliding force can be obtained in the TT � �TxT � +�TyT � +�TzT � . (15) directions of X-, Y-, and Z-axes (see Figure 10): Advances in Civil Engineering 9

+e direction cosines of TT with Cartesian coordinate Z axes are αT, βT, and cT. T +e vector angle (φc) between Ps and T can be de- P scribed in the following form (see Figure 11):

φc � arccos αsαT + βsβT + cscT�. (16)

γp +e stable coefficient in the X-axis direction is φc xT x T α β ( ) p β T FCSRM � . 17 P α X Pxs T γT +e stable coefficient in the Y-axis direction is TyT y Y FCSRM � . (18) Pys T +e stable coefficient in the Z-axis direction is TzT Figure z ( ) 11: Vector and relation diagram of glide force and antislip FCSRM � . 19 Pzs force in potentially destructive mode. +e stable coefficient in the sliding force direction is �� p 2 2 2 ( ) defined as T � �Txp � +�Typ � +�Tzp � . 24 TT s cos ϕc ( ) p FCRSM � . 20 +e direction cosines of T with Cartesian coordinate Ps axes are αr, βr, and cr. p +e vector angle (φm) between P and Fsf � TSF − Flf can be described in the following form: 7.1.2. Main ,rust Method (MTM). +e main thrust method αm � arccos�αpαr + βpβr + cpcr�. (25) is used only to evaluate the stability of a thrust-type landslide. +e critical state curves (see Figure 10, dashed line DE) +e stable coefficient in the X-axis direction is can be obtained, and the residual thrust force from the xp posterior region to the critical state curve (DE) can be x T FMTM � . (26) calculated (see Figure 10): Pxp

P � B � u + u − b − b� y z, +e stable coefficient in the Y-axis direction is xp f σθ στ σθ στ d d Ωx Typ P � B � u + u − b − b� x z, Fy � . yp f σθ στ σθ στ d d (21) MTM (27) Ωx Pyp P � B � u + u − b − b� x y. zp f σθ στ σθ στ d d Ωx +e stable coefficient in the Z-axis direction is p Tzp +e vector sum (P ) of Pxp,Pyp, and Pzp is Fz � . �� MTM (28) Pzp p 2 2 2 ( ) P � �Pxp � +�Pyp � +�Pzp � . 22 +e stable coefficient in the main slip force direction is p p +e direction cosines of P with Cartesian coordinate s T cos ϕm F � p . (29) axes are αp, βp, and cp. MTM P +e differential value (or residual frictional force) be- p,b p,b p,b tween the frictional force (σn , σθ , and στ ) under the b b b possible failure mode and the antislip force (σn, σθ, and στ) 7.1.3. Comprehensive Displacement Method (CDM). +e u u under the current situation can be obtained as follows: deformation from the present strain states (εθ and ετ ) is p,b calculated and is projected onto X-, Y-, and Z-axes: Txp � B �σp,b + σ + σp,b − σb − σb − σb�dydz, R n θ τ n θ τ Ωx u u yp p,b p,b p,b b b b Sxd � B ε + ε �dydz, T � B �σ + σ + σ − σ − σ − σ �dxdz, f R θ τ R n θ τ n θ τ (23) Ωx +Ωx Ωy S � B u + u � x z, zp p,b p,b p,b b b b yd f εθ ετ d d (30) T � B �σ + σ + σ − σ − σ − σ �dydx. Ω +ΩR R n θ τ n θ τ y y Ωz u u Szd � B εθ + ετ �dxdy. p x y z f+ R +e vector sum (T ) of Fp ,Fp , and Fp is Ωz Ωz 10 Advances in Civil Engineering

+e vector sum (αi) of ci is where Sxs,Sys, and Szs are the vector sums of the displace- �� ments in the directions of the X-, Y-, and Z-axes, respec- 2 2 2 S � S � +�S � +S � . (31) tively, and the vector sum (Ss) of Sxs,Sys, and Szs is d x d y d z d �� 2 2 2 ( ) +e direction cosines of F with Cartesian coordinate axes Ss � Sxs � +�Sys � +Szs � . 40 are σi . +e possible failure mode is analyzed, the distribution n p,b p,b p,b of the strain (εn , εθ , and ετ ) under the possible failure +e direction cosines of Ss with Cartesian coordinate mode can be calculated, and the vector sums of displacement axes are αs, βs, and cs. +e possible failure mode is analyzed, p.b p.b p.b under the possible failure mode can be obtained in the and the distribution of the strain (εn , εθ , and ετ ) under directions of X-, Y-, and Z-axes: the possible failure mode can be calculated. +e differences p.b p.b p.b b b b between εn , ε , and ε and ε , ε , and ε can be obtained p,b θ τ n θ τ Sxd � B �εp,b + ε + εp,b�dydz. f R n θ τ and projected in the directions of the X-, Y-, and Z-axes: Ω‘x+Ωx xs p,b p,b p,b b b b yd p,b p,b p,b S � B �εn + ε + ε − εn − ε − ε �dydz, S � B �ε + ε + ε �dxdz, k θ τ θ τ f R n θ τ (32) Ωx Ω‘y+Ωy ys p,b p,b p,b b b b S � B �ε + ε + ε − ε − ε − ε �dxdz, ( ) zd p,b p,b p,b k n θ τ n θ τ 41 S � B �ε + ε + ε �dydx. Ωy f R n θ τ Ω‘z+Ωz zs p,b p,b p,b b b b d x d y d z d S � B �εn + εθ + ετ − εn − εθ − ετ�dydx, +e vector sum (S ) of S ,S , and S is Ωk �� z 2 2 2 xs ys zs Sd � �Sx d � +�Sy d � +�Sz d � . (33) where S ,S , and S are the vector sums of the displacement differences in the directions of the X-, Y-, and Z-axes d between the possible failure mode and present status, re- +e direction cosines of S with Cartesian coordinate s xs ys zs d d d spectively. +e vector sum (S ) of S ,S , and S is axes are α , β , and c . +e vector angle (φd) between the �� Sd S vectors and d can be calculated as follows: Ss � Sxs �2 +Sys �2 +Szs �2. (42) d d d φd � arccos�α αd + β βd + c cd�. (34) +e direction cosines of Ss with Cartesian coordinate s s s +e stable coefficient in the X-axis direction is axes are α , β , and c . +e vector angle (φs) between the s vectors S and Ss can be calculated: Sxd Fx � . (35) s s s C DM φs � arccos α αs + β βs + c cs�. (43) Sxd +e stable coefficient in the Y-axis direction is +e stable coefficient in the X-axis direction is xs yd x S y S FS DM � . (44) FC DM � . (36) Ss Syd +e stable coefficient in the Y-axis direction is +e stable coefficient in the Z-axis direction is Sys zd Fy � . (45) z S S DM ( ) Ss FCDM � . 37 Szd +e stable coefficient in the Z-axis direction is +e stable coefficient in the slip displacement direction is zs z S s d d F � . ( ) F � S S . (38) S DM 46 C DM cos ϕd/ Ss +e stable coefficient in the main slip displacement 7.1.4. Surplus Displacement Method (SDM). +e strains direction is (εu and εu) from the posterior region to the critical stress Ss cos ϕ θ τ s s ( ) state (see Figure 10, DE (dashed line)) in the present status FS DM � . 47 Ss are calculated and are projected onto X-, Y-, and Z-axes:

S � B u + u � y z, xs f εθ ετ d d Ωx 7.2. Stability Analysis of a New Shear Stress Constitutive Model. According to the stability analysis method of the S � B u + u � x z, ys f εθ ετ d d (39) ideal elastoplastic constitutive model in Section 7.1, the Ω y stability coeﬃcient of the new shear stress constitutive S � B u + u � x y, models, such as the CSRM, the MTM, and the surplus zs f εθ ετ d d Ωz displacement method (SDM), can be obtained. Advances in Civil Engineering 11

7.3. Stability Coefficient Study 1 2 the m-th slice block 3 7.3.1. Some Term Definitions. Some terms are first defined. 4 f slope surface +e failure ratio ( r) is the value of the driving stress of rock m–1 or soil divided by its strength; when the value is greater than m m+1 1, the fr is equal to 1. +e failure percentage (fp) is the value n–1 n of the sum of the failure ratio multiplied by its area and sliding surface divided by the total area. +e failure area percentage (fs) is the value of the area sum corresponding to the failure ratio Figure 12: Schematic of the partial strength reduction method. (fr � 1) divided by the total area. +e frictional resistance variation coefficient (F ) is the f N � W cos α + P sinα − α � vector sum of the frictional resistance of the entire sliding i i i i− 1 i− 1 i body failure divided by the frictional resistance vector sum 1 2 1 2 1 during the progressive deformation and includes the X-, Y-, + cwhi,u sin αi − chi,b sin αi − �cihi,u + cihi,b�li. y and Z-axial (Fx ,F , and Fz ) and vector sum (F ) directions. 2 2 2 f f y f (49) +e variation coefficient of the driving sliding force (Fp) is the vector sum of the driving sliding force of the entire i Normal stress σn is sliding surface failure divided by the driving sliding force i N vector sum during the progressive deformation and includes σ � i. (50) x y z n l the X-, Y-, and Z-axial (Fp,Fp, and Fp) and vector sum (Fp) i directions. i Critical frictional stress τpeak is τi � c + σi tan φ . (51) 7.3.2. Comparison of Safety Factors. A partial strength re- peak i n i duction method is proposed, and its result is used to i Critical frictional resistance Tpeak is compare with the progressive failure analysis. +e steps of i the partial strength reduction method are as follows. Tpeak � cili + Ni tan φi. (52) First, the traditional safety factor (TSF) of the entire Ti sliding body is calculated by the traditional slice block Frictional force after strength reduction peak,F is i method. +en, the local safety factor (Flf) from the rear zone T to the critical state (Figure 12, where the m-th slice block is Ti � peak. (53) peak,F F assumed to be in a critical stress state under Flf � 1) is S obtained by the same method. +e surplus safety factor Driving sliding force Pi (P0 � 0) is (Fsf � TSF − Flf) is equal to the safety factor of the entire S sliding body minus the local safety factor. +is factor is used Pi � Wi sin αi + Pi− 1 cosαi− 1 − αi � to compare with that of the MTM during the progressive (54) 1 1 failure process. + c h2 cos α − c h2 cos α . 2 w i,u i 2 w i,b i Unbalanced thrust force P is 7.3.3. Unbalanced ,rust Method. +e unbalanced thrust i method (UTM) is taken as an example to calculate the safety S i Pi � Pi − Tpeak,F, (55) factor, and its formula is derived (Figure 13). +e fundamental assumptions are as follows: where Wi is the weight of the i-th slice block, li is the length of the i-th slice block bottom, α is the angle between the (1) +e slice block is assumed to have strong defor- i bottom and the horizontal axis of the i-th slice block, c is the mation capacity and is classified by a vertical interval i cohesion of the bottom of the i-th slice block, φi is the (2) +e former slice block force from the posterior slice frictional angle of the i-th slice block, F is the stability factor, block is parallel to the bottom side of the posterior i σn is the normal stress of the i-th slice block, cw is the specific slice block gravity of water, hi,u is the height of water at the left of the (3) +e rotation of the slice block is not considered slice block, and hi,b is the height of water level at the right of (4) +e frictional stress is satisfied with the new shear the slice block. stress-strain model +e relationship between the shear strain of two slice 8. Case Study blocks can be described in the failure zone as follows: +e progressive failure process is presented for the Kaziwan ci+1 landslide in the +ree Gorges Reservoir. ci � . (48) cosαi − αi+1 � In the i-th slice block, we have the following. 8.1. Geological Survey. +e Kaziwan landslide is situated in Normal pressure Ni is Zigui County, and its geographic coordinates are as follows: 12 Advances in Civil Engineering

Phreatic line

hr,l Ni Ti αi pi αi

pi-l hi,l αi-l

Wi γwhr,l

αi

li γwhi,l

Figure 13: +e force distribution of the i-th soil slice block.

(X: 3432550, Y: 37471500), longitude: 110°41′37″E, and Guizhou River latitude: 31° 0′48″N. +e Kaziwan landslide is located on the I left bank of the Guizou River, 1.9 km from the mouth and 44 km from the +ree Gorges Dam. Its basic characteristics are as follows. +e trailing edge elevation is 720 m, the leading edge elevation is 85 m, the slope aspect is 296°, the slope length is 1270 m, the slope height is 635 m, and the slope degree is 15° ∼40° (Figure 14). Quaternary deposits are distributed across the slope. +e deposits consist of broken stone and silty clay. Most stone is Quaternary deposits located at the toe of the slope; boulders can be seen in the Quaternary debris front, the particle sizes range from 0.4 m to 1.0 m, and the Quaternary landslides largest is more than 1.0 m. +e stratigraphic lithology is Boundary of Landslides interbedded sandstone and mudstone of the Upper Jurassic Figure Suining Formation. +e purple-red silty mudstone, siltstone, 14: Plan view of the Kaziwan landslide. and feldspar sandstone crop out the northeast of the slope. +e occurrence of bedrock is 280°∼320°∠26°∼39°, and the 1000 m; the terrain gradient at the boundary is gentle, and occurrences of ﬁssures are 180°∠82° and 240∼250°∠46∼79°. the gradient is approximately 20°. Its right boundary di- Cracks are ﬁlled with mudstone. +e lateral bedrock crops rection is 210°, and its length is 1000 m. +e landslide is out to the southwest, and its occurrence is 65°∠66°. +e composed of sandstone and mudstone, which are easily Kaziwan slope is downward. softened by water immersion to form a weak layer. +e +e Guizhou River is an open face of the Kaziwan sliding zone between the sandstone and mudstone is formed landslide. Its left boundary direction is 340°, and its length is naturally. Advances in Civil Engineering 13

1 phreatic line (50 years) 2 3 4 6 5 7 8 9 10 175m water level line 12 11 13 15 14 16 17 phreatic line (20 years) 19 18 22 21 20 24 23 26 25 28 27 30 29 32 31 34 33 3635

Figure 15: Block division map of the Kaziwan landslide.

Table 1: Safety factor of progressive failure under the rainfall with a � − . , recurrence interval of 20 years. ρi,0 0 9999 ρi,c � − 0.51, CSB 25 26 27 28 29 30 n,c TSF 1.317 1.349 1.399 1.410 1.428 1.484 σi � 900 kPa, Fx 0.746 0.704 0.655 0.607 0.559 0.510 ς � 1.28, yCSRM i FCSRM 2.925 2.808 2.667 2.526 2.385 2.244 s ai,3 � 0.0129, FCSRM 1.718 1.558 1.540 1.523 1.475 1.410 (56) x FMTM 0.202 0.193 0.175 0.157 0.124 0.108 ai,2 � 1500 kPa, y FMTM 0.205 0.197 0.186 0.172 0.147 0.121 s ai,1 � 2000 kPa, FMTM 0.203 0.195 0.189 0.166 0.135 0.111 x b � 50, FCDM 1.152 1.141 1.133 1.117 1.102 1.088 1 y FCDM 1.238 1.206 1.193 1.181 1.172 1.162 1 s b � 0� �. FCDM 1.188 1.175 1.161 1.157 1.144 1.133 2 x kPa FSDM 0.284 0.268 0.256 0.211 0.185 0.163 y FSDM 0.402 0.392 0.342 0.319 0.288 0.267 +e rainfall with a recurrence interval of 20 years is s FSDM 0.344 0.333 0.326 0.300 0.282 0.236 researched, and the seepage curve can be obtained by nu- CSB 31 32 33 34 35 36 merical analysis. +e TSF is 1.587. TSF 1.495 1.508 1.519 1.544 1.559 1.587 +e critical stress state is located in the 25th slice block, x FCSRM 0.462 0.414 0.365 0.317 0.269 0.220 y when the partial strength reduction method proposed in this FCSRM 2.103 1.962 1.821 1.680 1.539 1.398 s paper is used and the safety factor is equal to 1. When the FCSRM 1.320 1.290 1.214 1.117 1.049 1.011 x critical state block (CSB) moves forward step by step, the FMTM 0.097 0.075 0.043 0.024 0.018 0.00 y different safety factors are presented (Table 1) under the FMTM 0.101 0.092 0.067 0.043 0.037 0.00 s rainfall with a recurrence interval of 20 years. FMTM 0.099 0.084 0.046 0.029 0.021 0.00 x +e rainfall once in 50 years is also researched, and the FCDM 1.064 1.053 1.042 1.035 1.025 1.00 y TSF is 1.462. +e critical stress state is located at the 30th FCDM 1.142 1.121 1.101 1.090 1.070 1.00 s FCDM 1.120 1.110 1.075 1.055 1.020 1.00 slice block, when the partial strength reduction method x FSDM 0.142 0.109 0.081 0.048 0.026 0.00 proposed in this paper is used and the safety factor is equal to y FSDM 0.253 0.218 0.173 0.127 0.071 0.00 1. When the CSB moves forward step by step, the different s FSDM 0.215 0.193 0.122 0.101 0.054 0.00 safety factors are presented (see Table 2) under the rainfall +e normal deformation is neglected. once in 50 years. +e TSFs (1.587 and 1.462) are obtained under the rainfall once in 20 and 50 years, respectively. +e surplus 8.2. Calculation and Analysis. +e calculated slice block safety factors are 0.587 and 0.462, respectively, but the safety scheme (Figure 15) can be obtained along the profile I-I’ factors obtained by the MTM are 0.202, 0.203, 0.203 and (Figure 14). +e specific weight of the sliding body is 3 0.097, 0.101, 0.099 in the X-axial, Y-axial, and main sliding 20 kN/m , and the angle and length of the bottom margin of directions, respectively. +ese results show that the surplus the slice block can be seen in Figure 15. safety factors (0.587, 0.462) obtained by the traditional +e parameters of the model are listed as follows: method are greater than those (0.203, 0.099) of the new c � 24 kPa, method (MTM) proposed in this paper. Tables 1 and 2 show that the safety factor decreases while ϕ � 23∘, the critical stress state moves forward step by step. Finally, when the 36th slice block is in the critical stress state, the G � 2850 kPa, safety factors of MTM and SDM are equal to 0 and the safety 14 Advances in Civil Engineering

Table 2: Safety factor of progressive failure under once in 50 years pressure is equal to the counterpressure corresponding to of heavy rain. the critical state. +e driving sliding force and pressure are CSB 30 31 32 33 34 35 36 equal to the frictional resistance and counterpressure, respectively, in the stable and less-stable zones and at the TSF 1.21 1.344 1.407 1.415 1.433 1.447 1.462 Fx 0.595 0.527 0.469 0.394 0.340 0.264 0.210 critical state, but the frictional resistance reaches its maxi- yCSRM mum at the critical state and both shear stress and shear FCSRM 2.645 2.454 2.246 2.073 1.845 1.665 1.456 s strain are discontinuous in the failure zone. FCSRM 1.743 1.643 1.439 1.350 1.156 1.076 0.976 x FMTM 0.100 0.071 0.048 0.035 0.028 0.017 0.00 +e stability of the slope is directly related to the critical y FMTM 0.197 0.131 0.107 0.086 0.054 0.031 0.00 state. +e force and moment balance corresponding to the s FMTM 0.120 0.085 0.068 0.052 0.034 0.019 0.00 material strength are used to describe the critical state for the x FCDM 1.210 1.193 1.114 1.094 1.055 1.024 1.00 y slice method. Design methods for slope control are proposed FCDM 1.310 1.258 1.200 1.177 1.126 1.069 1.00 s according to the failure characteristics. In method I, a rigid FCDM 1.227 1.199 1.120 1.108 1.069 1.019 1.00 x design is suggested; the slope control position is selected at FSDM 0.140 0.107 0.079 0.067 0.041 0.033 0.00 y the critical state for the thrust- and pull-type slopes and at FSDM 0.271 0.217 0.171 0.128 0.077 0.039 0.00 Fs 0.191 0.161 0.129 0.080 0.055 0.036 0.00 the yield limit stress state for the foundation pit with a safety SDM factor. In method II, a flexible design is proposed; the slope +e normal deformation is neglected. control position is selected at the yield limit stress state for the thrust- and pull-type slopes and at the peak limit stress factor of CDM is equal to 1; thus, the entire Kaziwan state for the foundation pit with a safety factor. In method landslide maintains the critical state. +e physical signifi- III, a rigid-flexible design is proposed, and the slope control cance of the MTM, SDM, and CDM are very clear. position is selected between those of methods I and II with a safety factor. 9. Conclusion Some terms (failure ratio, etc.) are defined, and they are used for mechanical analysis and probability theory appli- +e failure rule for a slope is investigated, and several results cation to slope engineering. +e definition of the TSF (the are obtained in this paper. +e failure development direction maximum frictional resistance divided by the driving sliding is defined, and the mechanical failure mode can be deduced force) is worth discussing, and it is possible for the TSF to from the cracking trajectories of the slope. evaluate an element (or a slice block). +e research results Five failure modes are proposed for thrust-type slopes. In show that the MTM, CDM, and SDM are feasible for mode I, shear failure occurs. In mode II, tensile (or tensile- evaluating the stability of a slope with regressive failure. shear) failure occurs in the rear zone and shear failure occurs in the other zones. In mode III, tensile (or tensile-shear) Data Availability failure occurs in the front zone and shear failure occurs in the other zones. In mode IV, the mechanical failure mode is +e data used to support the findings of this study are in- a combination of modes II and III. In mode V, mechanical cluded within the article. failure occurs by alternating shear-tensile-shear modes. +at the failure occurs is defined along the sliding surface. Conflicts of Interest For pull-type slopes, in mode I, shear failure occurs along the whole sliding surface. In mode II, shear failure occurs in +e authors declare that they have no conflicts of interest. the front zone and tensile (or tensile-shear) failure occurs in the rear zone. In mode III, shear failure occurs in the front Acknowledgments zone and tensile failure occurs in the rear zone of the sliding body. Mode IV corresponds to a rock mass with distributed +is work was supported by the National Natural Science joints (or fissures): shear, tensile, and shear failures occur Foundation of China (Grant nos. 42071264 and 41372363). alternately along the soft interlayer and joints (or fissures). +is work was also supported by the Sichuan Science and Mode V is a combination of mode IV and a shear failure (or Technology Program (Grant no. 2019JDJQ0018) and sup- tensile-shear or tensile failure). ported by the Fundamental Research Funds for the Central Five existing forms in the field are proposed for slopes. In Universities (Grant no. 2682018GJ02). form I, the stress distribution is within the yield limit stress state. In form II, the various states of elastic, elastoplastic, References and peak stresses are distributed in different zones. In form III, the states of previous peak, peak, postfailure, and residual [1] G.-L. Feng, X.-T. Feng, B.-r. Chen, Y.-X. Xiao, and Y. 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